Keys/Frame.lean — FrcCore

1import FrcCore.FrameCore2import FrcCore.Frame3import FrcCore.Parity4import FrcCore.Transform5import FrcCore.Orbit6import FrcCore.Instances7import FrcCore.Sum8import FrcCore.Meridian9import FrcCore.Theme.Domain1011/-!12# FrcCore.Keys.Frame — the keyed theorems of the frame theme (ledger migration, task LM19)1314One declaration per key, `FRC.Ledger.p<key>`, generated by `make_keys.py` from the ledgers' Lean bindings: the theorem15that every row carrying the key asserts, proved from the themes alone. The paper's module keeps its declaration16`FRC.<Module>.p<key>` as an alias. Every declaration has no axioms (`check_core_axioms.py`).17-/1819namespace FRC.Ledger2021-- Keys of the frame theme (generated by make_keys.py from the ledgers' Lean bindings; edit the ledgers, not this file)22/-- p00032 — 00:C12. \textbf{Temperature and the classical arity, derived}: the temperature domain is acceleration, $[\Theta]=[E][k_B]^{-1}=[L][T]^{-2}$, flag-free, $\Theta_P=E_P/|k_B|$, Unruh closing as a domain identity; neither mass nor temperature is primitive --- $M$-$L$-$T$-plus-thermal is the torsion-free shadow of the two-generators-plus-flag system (the mass primitive standing in for the flag, the thermal for $k_B$), classical dimensional analysis recovered in exponents and arity. -/23theorem p00032 : (FRC.Domain.Lift.temp = FRC.Domain.Lift.accel ∧ FRC.Domain.Lift.temp = { r := (1 : Int), s := (-2 : Int), j := (0 : Int) } ∧ FRC.Domain.Lift.kB = { r := (-1 : Int), s := (1 : Int), j := (1 : Int) }) ∧ (FRC.Domain.Lift.temp.j = (0 : Int) ∧ FRC.Domain.Lift.energy.j = (1 : Int) ∧ FRC.Domain.Lift.hbar.j = (1 : Int) ∧ FRC.Domain.Lift.kB.j = (1 : Int)) ∧ (FRC.Domain.Lift.unruh = FRC.Domain.Lift.temp ∧ FRC.Domain.Lift.unruh.j = (0 : Int)) ∧ (FRC.Domain.Lift.mass = { r := (-2 : Int), s := (1 : Int), j := (1 : Int) } ∧ FRC.Domain.Lift.temp = FRC.Domain.Lift.L * FRC.Domain.Lift.T⁻¹ * FRC.Domain.Lift.T⁻¹ ∧ FRC.Domain.Lift.energy * (FRC.Domain.Lift.hbar * FRC.Domain.Lift.speed⁻¹)⁻¹ = FRC.Domain.Lift.accel ∧ (FRC.Domain.Lift.energy * (FRC.Domain.Lift.hbar * FRC.Domain.Lift.speed⁻¹)⁻¹).j = (0 : Int)) ∧ (FRC.Domain.Lift.classical (1 : Int) (0 : Int) (0 : Int) (0 : Int) = FRC.Domain.Lift.mass ∧ FRC.Domain.Lift.classical (0 : Int) (1 : Int) (-1 : Int) (0 : Int) = FRC.Domain.Lift.speed ∧ FRC.Domain.Lift.classical (0 : Int) (1 : Int) (-2 : Int) (0 : Int) = FRC.Domain.Lift.accel ∧ FRC.Domain.Lift.classical (1 : Int) (2 : Int) (-2 : Int) (0 : Int) = FRC.Domain.Lift.energy ∧ FRC.Domain.Lift.classical (1 : Int) (2 : Int) (-1 : Int) (0 : Int) = FRC.Domain.Lift.hbar ∧ FRC.Domain.Lift.classical (1 : Int) (1 : Int) (-2 : Int) (0 : Int) = FRC.Domain.Lift.force ∧ FRC.Domain.Lift.classical (1 : Int) (1 : Int) (-1 : Int) (0 : Int) = FRC.Domain.Lift.momentum ∧ FRC.Domain.Lift.classical (1 : Int) (2 : Int) (-3 : Int) (0 : Int) = FRC.Domain.Lift.power ∧ FRC.Domain.Lift.classical (1 : Int) (-1 : Int) (-2 : Int) (0 : Int) = FRC.Domain.Lift.pressure ∧ FRC.Domain.Lift.classical (-1 : Int) (3 : Int) (-2 : Int) (0 : Int) = FRC.Domain.Lift.G ∧ FRC.Domain.Lift.classical (1 : Int) (2 : Int) (-2 : Int) (-1 : Int) = FRC.Domain.Lift.kB ∧ FRC.Domain.Lift.classical (0 : Int) (0 : Int) (0 : Int) (1 : Int) = FRC.Domain.Lift.temp) ∧ (FRC.Domain.Lift.classical (0 : Int) (1 : Int) (0 : Int) (0 : Int) = FRC.Domain.Lift.L ∧ FRC.Domain.Lift.classical (0 : Int) (0 : Int) (1 : Int) (0 : Int) = FRC.Domain.Lift.T ∧ FRC.Domain.Lift.classical (1 : Int) (2 : Int) (-1 : Int) (0 : Int) = FRC.Domain.Lift.flag ∧ FRC.Domain.Lift.classical (0 : Int) (0 : Int) (0 : Int) (1 : Int) = FRC.Domain.Lift.classical (0 : Int) (1 : Int) (-2 : Int) (0 : Int) ∧ FRC.Domain.Lift.classical (0 : Int) (-1 : Int) (2 : Int) (1 : Int) = (1 : FRC.Domain.Lift) ∧ (FRC.Domain.Lift.classical (1 : Int) (0 : Int) (0 : Int) (0 : Int)).r * ((FRC.Domain.Lift.classical (0 : Int) (1 : Int) (0 : Int) (0 : Int)).s * (FRC.Domain.Lift.classical (0 : Int) (0 : Int) (1 : Int) (0 : Int)).j - (FRC.Domain.Lift.classical (0 : Int) (1 : Int) (0 : Int) (0 : Int)).j * (FRC.Domain.Lift.classical (0 : Int) (0 : Int) (1 : Int) (0 : Int)).s) - (FRC.Domain.Lift.classical (1 : Int) (0 : Int) (0 : Int) (0 : Int)).s * ((FRC.Domain.Lift.classical (0 : Int) (1 : Int) (0 : Int) (0 : Int)).r * (FRC.Domain.Lift.classical (0 : Int) (0 : Int) (1 : Int) (0 : Int)).j - (FRC.Domain.Lift.classical (0 : Int) (1 : Int) (0 : Int) (0 : Int)).j * (FRC.Domain.Lift.classical (0 : Int) (0 : Int) (1 : Int) (0 : Int)).r) + (FRC.Domain.Lift.classical (1 : Int) (0 : Int) (0 : Int) (0 : Int)).j * ((FRC.Domain.Lift.classical (0 : Int) (1 : Int) (0 : Int) (0 : Int)).r * (FRC.Domain.Lift.classical (0 : Int) (0 : Int) (1 : Int) (0 : Int)).s - (FRC.Domain.Lift.classical (0 : Int) (1 : Int) (0 : Int) (0 : Int)).s * (FRC.Domain.Lift.classical (0 : Int) (0 : Int) (1 : Int) (0 : Int)).r) = (1 : Int) ∧ FRC.Domain.Lift.flag * FRC.Domain.Lift.flag * FRC.Domain.Lift.flag * FRC.Domain.Lift.flag ≠ (1 : FRC.Domain.Lift)) ∧ FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) FRC.Domain.Lift.temp = FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) FRC.Domain.Lift.accel ∧ FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) FRC.Domain.Lift.unruh = FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) FRC.Domain.Lift.temp ∧ FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) FRC.Domain.Lift.temp = ((1 : Int), (10 : Int)) ∧ FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) FRC.Domain.Lift.flag = ((0 : Int), (3 : Int)) ∧ FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) (FRC.Domain.Lift.T * FRC.Domain.Lift.T * FRC.Domain.Lift.T) = ((0 : Int), (3 : Int)) ∧ FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) (FRC.Domain.Lift.flag * FRC.Domain.Lift.flag * FRC.Domain.Lift.flag * FRC.Domain.Lift.flag) = ((0 : Int), (0 : Int)) ∧ FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) FRC.Domain.Lift.flag ≠ ((0 : Int), (0 : Int)) ∧ FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) (FRC.Domain.Lift.flag * FRC.Domain.Lift.flag) ≠ ((0 : Int), (0 : Int)) ∧ FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) (FRC.Domain.Lift.flag * FRC.Domain.Lift.flag * FRC.Domain.Lift.flag) ≠ ((0 : Int), (0 : Int)) ∧ FRC.Domain.Lift.reduce (13 : Nat) (3 : Nat) FRC.Domain.Lift.mass = ((11 : Int), (4 : Int)) ∧ FRC.Domain.Lift.reduce (173 : Nat) (43 : Nat) FRC.Domain.Lift.temp = FRC.Domain.Lift.reduce (173 : Nat) (43 : Nat) FRC.Domain.Lift.accel ∧ FRC.Domain.Lift.reduce (173 : Nat) (43 : Nat) FRC.Domain.Lift.unruh = FRC.Domain.Lift.reduce (173 : Nat) (43 : Nat) FRC.Domain.Lift.temp ∧ FRC.Domain.Lift.reduce (173 : Nat) (43 : Nat) FRC.Domain.Lift.temp = ((1 : Int), (170 : Int)) ∧ FRC.Domain.Lift.reduce (173 : Nat) (43 : Nat) FRC.Domain.Lift.flag = ((0 : Int), (43 : Int)) ∧ FRC.Domain.Lift.reduce (173 : Nat) (43 : Nat) (FRC.Domain.Lift.flag * FRC.Domain.Lift.flag * FRC.Domain.Lift.flag * FRC.Domain.Lift.flag) = ((0 : Int), (0 : Int)) ∧ FRC.Domain.Lift.reduce (173 : Nat) (43 : Nat) (FRC.Domain.Lift.flag * FRC.Domain.Lift.flag) = ((0 : Int), (86 : Int)) ∧ FRC.Domain.Lift.reduce (173 : Nat) (43 : Nat) (FRC.Domain.Lift.flag * FRC.Domain.Lift.flag * FRC.Domain.Lift.flag) = ((0 : Int), (129 : Int)) ∧ FRC.Domain.Lift.reduce (173 : Nat) (43 : Nat) FRC.Domain.Lift.mass = ((171 : Int), (44 : Int)) ∧ FRC.Domain.Lift.reduce (1373 : Nat) (343 : Nat) FRC.Domain.Lift.temp = FRC.Domain.Lift.reduce (1373 : Nat) (343 : Nat) FRC.Domain.Lift.accel ∧ FRC.Domain.Lift.reduce (1373 : Nat) (343 : Nat) FRC.Domain.Lift.unruh = FRC.Domain.Lift.reduce (1373 : Nat) (343 : Nat) FRC.Domain.Lift.temp ∧ FRC.Domain.Lift.reduce (1373 : Nat) (343 : Nat) FRC.Domain.Lift.flag = ((0 : Int), (343 : Int)) ∧ FRC.Domain.Lift.reduce (1373 : Nat) (343 : Nat) (FRC.Domain.Lift.flag * FRC.Domain.Lift.flag * FRC.Domain.Lift.flag * FRC.Domain.Lift.flag) = ((0 : Int), (0 : Int)) ∧ FRC.Domain.Lift.reduce (1373 : Nat) (343 : Nat) (FRC.Domain.Lift.flag * FRC.Domain.Lift.flag) = ((0 : Int), (686 : Int)) ∧ FRC.Domain.Lift.reduce (1373 : Nat) (343 : Nat) (FRC.Domain.Lift.flag * FRC.Domain.Lift.flag * FRC.Domain.Lift.flag) = ((0 : Int), (1029 : Int)) ∧ FRC.Domain.Lift.reduce (1373 : Nat) (343 : Nat) FRC.Domain.Lift.mass = ((1371 : Int), (344 : Int)) :=24  And.intro @FRC.Domain.Lift.temp_eq_accel (And.intro @FRC.Domain.Lift.temp_flag_free (And.intro @FRC.Domain.Lift.unruh_closes (And.intro @FRC.Domain.Lift.not_primitive (And.intro @FRC.Domain.Lift.classical_exponents (And.intro @FRC.Domain.Lift.classical_arity (@FRC.Domain.Lift.modular_shadow))))))25/-- p01004 — 1:B2. Symmetry completeness (Thm.~\ref{thm:symmetric-completeness}): the fourth roots of unity form the unique order-four subgroup $Q_4=\{1,\im,-1,-\im\}$; under the Klein four-group $\langle x\mapsto-x,\,x\mapsto x^{-1}\rangle$, $Q_4$ is the union of the two size-two orbits $\{\pm1\}$, $\{\pm\im\}$, and $\Fpx\setminus Q_4$ splits into exactly $\kap-1$ orbits of size four. For $\p\equiv3\pmod4$ no $\im$ exists. -/26theorem p01004 : (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∃ u, u * u = (-1 : FRC.Shell p)) ∧ (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (x : FRC.Shell p), x ^ (4 : Nat) = (1 : FRC.Shell p) ↔ x = (1 : FRC.Shell p) ∨ x = (-1 : FRC.Shell p) ∨ x = FRC.Shell.Frame.quarterTurn g κ ∨ x = -FRC.Shell.Frame.quarterTurn g κ) ∧ (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {x y : FRC.Shell p}, x * y = (1 : FRC.Shell p) → x ^ (4 : Nat) ≠ (1 : FRC.Shell p) → x ≠ -x ∧ x ≠ y ∧ x ≠ -y ∧ -x ≠ y ∧ -x ≠ -y ∧ y ≠ -y) ∧ (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {x : FRC.Shell p}, x ≠ (0 : FRC.Shell p) → x ^ (4 : Nat) ≠ (1 : FRC.Shell p) → ∃ r, (1 : Nat) ≤ r ∧ r < κ ∧ FRC.Shell.Frame.InOrbit (g ^ r) x) ∧ (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {r r' : Nat}, (1 : Nat) ≤ r → r < κ → (1 : Nat) ≤ r' → r' < κ → ∀ {x : FRC.Shell p}, FRC.Shell.Frame.InOrbit (g ^ r) x → FRC.Shell.Frame.InOrbit (g ^ r') x → r = r') ∧ (∀ (x : Fin (13 : Nat)), ↑x = (1 : Nat) ∨ ↑x = (5 : Nat) ∨ ↑x = (12 : Nat) ∨ ↑x = (8 : Nat) ↔ FRC.Shell.ofNat ↑x ^ (4 : Nat) = (1 : FRC.Shell (13 : Nat))) ∧ (2 : FRC.Shell (13 : Nat)) * (7 : FRC.Shell (13 : Nat)) = (1 : FRC.Shell (13 : Nat)) ∧ (3 : FRC.Shell (13 : Nat)) * (9 : FRC.Shell (13 : Nat)) = (1 : FRC.Shell (13 : Nat)) :=27  And.intro @FRC.Shell.Frame.quarter_turn_exists (And.intro @FRC.Shell.Frame.fourth_roots (And.intro @FRC.Shell.Frame.klein_orbit_four (And.intro @FRC.Shell.Frame.orbit_rep (And.intro @FRC.Shell.Frame.orbit_rep_unique (@FRC.Shell.s13_klein)))))28/-- p01005 — 1:B3. The oriented quarter-turn: $\im=-\gen^{\kap}$ satisfies $\im^{2}=-1$ for every primitive root $\gen$; the two square roots of $-1$ are $\{-\gen^{\kap},\gen^{\kap}\}$, so the drive fixes which is $\im$. On $\F_{13}$, $\gen=2$: $\im=5$. -/29theorem p01005 : (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → FRC.Shell.Frame.quarterTurn g κ * FRC.Shell.Frame.quarterTurn g κ = (-1 : FRC.Shell p)) ∧ FRC.Shell.Frame.quarterTurn (2 : FRC.Shell (13 : Nat)) (3 : Nat) = (5 : FRC.Shell (13 : Nat)) ∧ (5 : FRC.Shell (13 : Nat)) * (5 : FRC.Shell (13 : Nat)) = (-1 : FRC.Shell (13 : Nat)) ∧ (-5 : FRC.Shell (13 : Nat)) = (8 : FRC.Shell (13 : Nat)) ∧ (2 : FRC.Shell (13 : Nat)) ^ (6 : Nat) = (-1 : FRC.Shell (13 : Nat)) ∧ (2 : FRC.Shell (13 : Nat)) ^ (5 : Nat) = (6 : FRC.Shell (13 : Nat)) :=30  And.intro @FRC.Shell.Frame.quarter_turn_sq (@FRC.Shell.s13_datum)31/-- p01006 — 1:B4. Affine reframing (Def.~\ref{def:framed-field}, Lemma~\ref{lem:affine-invariance}): $\varphi_{a,b}(x)=a+bx$, $b\neq0$, is a ring isomorphism $(\Fp,+,\cdot)\to(\Fp,\oplus,\otimes)$ with origin $a$ and unit $a+b$; every polynomial identity is frame-covariant. -/32theorem p01006 : ∀ {p : Nat} [FRC.Pos p] {a b y : FRC.Shell p}, b * y = (1 : FRC.Shell p) → ∀ (z : FRC.Shell p), FRC.Shell.Frame.affineMul a b y (a + b) z = z :=33  @FRC.Shell.Frame.affine_frame_unit34/-- p01011 — 1:C4. Involutions and counts (Def.~\ref{def:orbit-sphere}\,(a)--(c)): $M_n(a)=M_{n+2\kap}(-a)$ and $L_a(m)=L_{-a}(m+2\kap)$ since $\gen^{2\kap}=-1$; $\p-1$ distinct meridian lists in $2\kap$ great circles; $2\kap$ latitude pairs; one-sided vertex count $(\p-1)^{2}/2+1$, the double cover $(\p-1)^{2}$ (00:B11). -/35theorem p01011 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (a : FRC.Shell p) (n : Nat), -a * g ^ (n + (2 : Nat) * κ) = a * g ^ n :=36  @FRC.Shell.Frame.meridian_involution37set_option linter.defProp false in38/-- p01013 — 1:D2. The window law: $z\mapsto z\bmod\p$ is injective on $W_H$ when $2H<\p$; for $x,y\in W_H$ the residue of $x\pm y$ reads back the integer when $4H<\p$, and that of $xy$ when $2H^{2}<\p$ (the same injectivity on the windows $2H$ and $H^{2}$). -/39def p01013 := And.intro @FRC.Shell.Frame.window_injective (And.intro @FRC.Shell.Frame.window_signed (And.intro @FRC.Shell.Frame.window_add_readback (@FRC.Shell.Frame.window_mul_readback)))40/-- p01015 — 1:D4. Scale-periodicity (Lemma~\ref{thm:scale-periodicity}, in the field): $S_n(x)=x\gen^{-n}$ satisfies $S_{n+(\p-1)}=S_n$; the grids $G_n$ are $(\p-1)$-periodic as ordered lists; the zoom $[x,n]\mapsto[x,n+1]$ has order $\p-1$. As lists of rationals $x/\gen^{n}$ ($\gen$ lifted to $\Z$) the grids are not periodic: the periodicity is a relation of the field. -/41theorem p01015 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (x : FRC.Shell p) (n : Nat), x * g ^ (n + (p - (1 : Nat))) = x * g ^ n :=42  @FRC.Shell.Frame.scale_periodic43/-- p01019 — 1:E2. The extension by $X^{2}+1$ is trivial (Prop.~\ref{prop:Cp-field} reversed): on the shell $\Fp[X]/(X^{2}+1)\cong\Fp\times\Fp$ has zero divisors, $(X+\im)(X-\im)=0$, so it is not a field; $\Fp[X]/(X^{2}+1)$ is a field exactly when $\p\equiv3\pmod4$; the quadratic extension of the shell is $\Fp[\eta]$, $\eta^{2}=\nu$ a non-square (8-dirac B6). -/44theorem p01019 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → FRC.Shell.Frame.cmul (FRC.Shell.Frame.quarterTurn g κ, (1 : FRC.Shell p)) (-FRC.Shell.Frame.quarterTurn g κ, (1 : FRC.Shell p)) = ((0 : FRC.Shell p), (0 : FRC.Shell p)) ∧ (FRC.Shell.Frame.quarterTurn g κ, (1 : FRC.Shell p)) ≠ ((0 : FRC.Shell p), (0 : FRC.Shell p)) ∧ (-FRC.Shell.Frame.quarterTurn g κ, (1 : FRC.Shell p)) ≠ ((0 : FRC.Shell p), (0 : FRC.Shell p)) :=45  @FRC.Shell.Frame.complex_chart_zero_divisor46/-- p01020 — 1:Z1. No element of additive order two (Thm.~\ref{thm:no-south-pole}): $2s=0$ forces $s=0$; the antipode of the origin on the additive cycle is not a residue; it sits between $2\kap=(\p-1)/2$ and $2\kap+1=(\p+1)/2=2^{-1}$ (20-rh B8). -/47theorem p01020 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (s : FRC.Shell p), (2 : FRC.Shell p) * s = (0 : FRC.Shell p) → s = (0 : FRC.Shell p) :=48  @FRC.Shell.Frame.no_south_pole49/-- p02007 — 2:B3. Primitive generators form one orbit: every generator is $\gen^{u}$ with $u\in\Zp^{\times}$, a torsor for $\operatorname{Aut}(C_{\p-1})$ (Props.~\ref{prop:primitive-orbit}, \ref{prop:no-canonical-e}). -/50theorem p02007 : (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (h : FRC.Shell p), h.IsPrimitive (p - (1 : Nat)) ↔ ∃ u, u < p - (1 : Nat) ∧ FRC.Shell.Coprime u (p - (1 : Nat)) ∧ h = g ^ u) ∧ FRC.Shell.Coprime (5 : Nat) (12 : Nat) ∧ FRC.Shell.Coprime (7 : Nat) (12 : Nat) ∧ FRC.Shell.Coprime (11 : Nat) (12 : Nat) ∧ ¬FRC.Shell.Coprime (2 : Nat) (12 : Nat) ∧ FRC.Shell.IsPrimitive (6 : FRC.Shell (13 : Nat)) (12 : Nat) ∧ ¬FRC.Shell.IsPrimitive (4 : FRC.Shell (13 : Nat)) (12 : Nat) ∧ (2 : FRC.Shell (13 : Nat)) ^ (5 : Nat) = (6 : FRC.Shell (13 : Nat)) :=51  And.intro @FRC.Shell.Frame.generator_orbit (@FRC.Shell.s13_orbit)52/-- p02013 — 2:D1. The half-period: $\gen^{\pi}=-1$ for every primitive $\gen$, $\pi=2\kap$ (Prop.~\ref{prop:half-period}); $2\pi\equiv-1$ (00:C1). -/53theorem p02013 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → g ^ ((2 : Nat) * κ) = (-1 : FRC.Shell p) :=54  @FRC.Shell.Frame.half_period55/-- p02014 — 2:D2. The quarter-turn subgroup $\Qp=\{x:x^{4}=1\}$ is the unique subgroup of order four; $\gen^{\kap}$ has order four with $(\gen^{\kap})^{2}=-1$; $\Qp=\{\pm1,\pm\im\}$ (Def.~\ref{def:quarter-turn-subgroup}, Prop.~\ref{prop:quarter-turn}; 1:B2). -/56theorem p02014 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → (g ^ κ) ^ (2 : Nat) = (-1 : FRC.Shell p) ∧ (g ^ κ) ^ (4 : Nat) = (1 : FRC.Shell p) :=57  @FRC.Shell.Frame.quarter_turn_order58/-- p02015 — 2:D3. The oriented quarter-turn $\im=-\gen^{\kap}=\gen^{-\kap}$, $\im^{2}=-1$, orientation by A2 (Rem.~\ref{rem:observer-canonical-e}); on $\F_{13}(\chron;0,1,2)$: $\im=5$, $-\im=8$, $\pi=6$ [value] (Fig.~\ref{fig:f13-constant-datum}). -/59theorem p02015 : FRC.Shell.Frame.quarterTurn (2 : FRC.Shell (13 : Nat)) (3 : Nat) = (5 : FRC.Shell (13 : Nat)) ∧ (5 : FRC.Shell (13 : Nat)) * (5 : FRC.Shell (13 : Nat)) = (-1 : FRC.Shell (13 : Nat)) ∧ (-5 : FRC.Shell (13 : Nat)) = (8 : FRC.Shell (13 : Nat)) ∧ (2 : FRC.Shell (13 : Nat)) ^ (6 : Nat) = (-1 : FRC.Shell (13 : Nat)) ∧ (2 : FRC.Shell (13 : Nat)) ^ (5 : Nat) = (6 : FRC.Shell (13 : Nat)) :=60  @FRC.Shell.s13_datum61/-- p02017 — 2:D5. Orientation classes: for $\gen'=\gen^{u}$, $\im'=-\gen'^{\kap}$ equals $\im$ when $u\equiv1\pmod4$ and $-\im$ when $u\equiv3\pmod4$; the oriented quarter-turn is frame-covariant within one class of generators, not under every generator change. -/62theorem p02017 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (u : Nat), (u % (4 : Nat) = (1 : Nat) → -(g ^ u) ^ κ = -g ^ κ) ∧ (u % (4 : Nat) = (3 : Nat) → -(g ^ u) ^ κ = - -g ^ κ) :=63  @FRC.Shell.Frame.orientation_class64/-- p02018 — 2:D6. The Euler identity on the shell: $\eu^{\im\pi}=\gen^{2\kap\im^{2}}=(-1)^{\im}$, equal to $-1$ exactly when the representative of $\im$ is odd (00:C14); the oriented frames of a shell are those with $-\gen^{\kap}$ odd --- on $\F_{13}$, $\gen=2$ ($\im=5$) is one, on $\F_{17}$, $\gen=3$ ($\im=4$) is not. -/65theorem p02018 : (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (i : Nat), (g ^ i) ^ (i * ((2 : Nat) * κ)) = if i % (2 : Nat) = (0 : Nat) then (1 : FRC.Shell p) else (-1 : FRC.Shell p)) ∧ FRC.Shell.Frame.quarterTurn (3 : FRC.Shell (17 : Nat)) (4 : Nat) = (4 : FRC.Shell (17 : Nat)) ∧ FRC.Shell.Frame.quarterTurn (6 : FRC.Shell (17 : Nat)) (4 : Nat) = (13 : FRC.Shell (17 : Nat)) ∧ ((3 : FRC.Shell (17 : Nat)) ^ (4 : Nat)) ^ ((4 : Nat) * (8 : Nat)) = (1 : FRC.Shell (17 : Nat)) ∧ ((6 : FRC.Shell (17 : Nat)) ^ (13 : Nat)) ^ ((13 : Nat) * (8 : Nat)) = (-1 : FRC.Shell (17 : Nat)) ∧ (3 : FRC.Shell (17 : Nat)) ^ (15 : Nat) = (6 : FRC.Shell (17 : Nat)) :=66  And.intro @FRC.Shell.Frame.euler_identity (@FRC.Shell.s17_orientation)67/-- p02019 — 2:D7. Negation $R(x)=-x$ and inversion $I(x)=x^{-1}$ are commuting involutions; off $\Qp$ the $\langle R,I\rangle$-orbit $\{x,-x,x^{-1},-x^{-1}\}$ has four elements, and $\Fpx\setminus\Qp$ splits into $\kap-1$ such orbits (Props.~\ref{prop:shell-negation}, \ref{prop:k4-shell}; 1:B2). -/68theorem p02019 : (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {x y : FRC.Shell p}, x * y = (1 : FRC.Shell p) → x ^ (4 : Nat) ≠ (1 : FRC.Shell p) → x ≠ -x ∧ x ≠ y ∧ x ≠ -y ∧ -x ≠ y ∧ -x ≠ -y ∧ y ≠ -y) ∧ (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {x : FRC.Shell p}, x ≠ (0 : FRC.Shell p) → x ^ (4 : Nat) ≠ (1 : FRC.Shell p) → ∃ r, (1 : Nat) ≤ r ∧ r < κ ∧ FRC.Shell.Frame.InOrbit (g ^ r) x) ∧ ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {r r' : Nat}, (1 : Nat) ≤ r → r < κ → (1 : Nat) ≤ r' → r' < κ → ∀ {x : FRC.Shell p}, FRC.Shell.Frame.InOrbit (g ^ r) x → FRC.Shell.Frame.InOrbit (g ^ r') x → r = r' :=69  And.intro @FRC.Shell.Frame.klein_orbit_four (And.intro @FRC.Shell.Frame.orbit_rep (@FRC.Shell.Frame.orbit_rep_unique))70/-- p02025 — 2:F1. $\gen$ is a principal $n$-th root of unity, $n=\p-1$: $\gen^{n}=1$, $\sum_{j<n}\gen^{jk}=0$ for $0<k<n$, and $n^{-1}=-1$ in $\Fp$ (Prop.~\ref{prop:principal-root}; $2\pi\equiv-1$). -/71theorem p02025 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (k : Nat), (0 : Nat) < k → k < p - (1 : Nat) → g ^ (p - (1 : Nat)) = (1 : FRC.Shell p) ∧ FRC.Shell.sumRange (fun j => (g ^ k) ^ j) (p - (1 : Nat)) = (0 : FRC.Shell p) ∧ FRC.Shell.ofNat (p - (1 : Nat)) = (-1 : FRC.Shell p) :=72  @FRC.Shell.Frame.principal_root73/-- p02027 — 2:F3. Inversion: $v_{j}=-\sum_{k<n}\mathcal F_{\gen}(v)_{k}\gen^{-jk}$, i.e.\ $W_{\gen}^{-1}=-(\gen^{-jk})_{jk}=-W_{\gen}J$ (Prop.~\ref{prop:ring-dft-inverse}; 6:B5), exact on every shell. -/74theorem p02027 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {k j : Nat}, k < p - (1 : Nat) → j < p - (1 : Nat) → FRC.Shell.sumRange (fun l => g ^ (l * k) * -g ^ ((p - (1 : Nat) - j) * l)) (p - (1 : Nat)) = if k = j then (1 : FRC.Shell p) else (0 : FRC.Shell p) :=75  @FRC.Shell.Frame.dft_inverse76/-- p02028 — 2:F4. Polynomial reading: $\mathcal F_{\gen}(v)_{k}=P_{v}(\gen^{k})$ with $P_{v}=\sum_{j}v_{j}X^{j}$; the transform exchanges coefficients with values on the phase cycle (Prop.~\ref{prop:polynomial-form}). -/77theorem p02028 : ∀ {p : Nat} [FRC.Pos p] (g : FRC.Shell p) (n : Nat) (v : Nat → FRC.Shell p) (k : Nat), FRC.Shell.Frame.dft g n v k = FRC.Shell.Frame.polyEval n v (g ^ k) :=78  @FRC.Shell.Frame.dft_eq_polyEval79/-- p02029 — 2:F5. Covariance under generator change: for $\gen'=\gen^{u}$, $v^{(\gen')}_{j}=v^{(\gen)}_{uj}$ and $\mathcal F_{\gen'}(v^{(\gen')})=\mathcal F_{\gen}(v^{(\gen)})$ (Prop.~\ref{prop:ring-dft-covariance}). -/80theorem p02029 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {u : Nat}, FRC.Shell.Coprime u (p - (1 : Nat)) → ∀ (v : Nat → FRC.Shell p) (k : Nat), FRC.Shell.Frame.dft (g ^ u) (p - (1 : Nat)) (fun j => v (u * j % (p - (1 : Nat)))) k = FRC.Shell.Frame.dft g (p - (1 : Nat)) v k :=81  @FRC.Shell.Frame.dft_covariance82/-- p06009 — 6:B2. The Euler identity is a property of the chart class: $e^{\im\pi}=\gen^{2\kap(\im\bmod2)}\equiv-1$ exactly when the quarter-turn residue is odd; the conjugate chart $(\gen,\im)\mapsto(\gen^{-1},-\im)$ toggles the parity, so one member of each conjugate pair carries it; on $\F_{13}(\chron;0,1,2)$, $\im=5$ and $6^{6}=-1$. Every primitive frame of the six shells. -/83theorem p06009 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (i : Nat), (g ^ i) ^ (i * ((2 : Nat) * κ)) = if i % (2 : Nat) = (0 : Nat) then (1 : FRC.Shell p) else (-1 : FRC.Shell p) :=84  @FRC.Shell.Frame.euler_identity85/-- p06010 — 6:B3. Chart covariance: under $\gen'=\gen^{u}$, $u\in\Zp^{\times}$, the quarter-turn name flips exactly on $u\equiv3\pmod4$ and the exponential-unit name moves already at $u\equiv1\pmod4$ whenever $\im(u-1)\not\equiv0\pmod{4\kap}$ ($\p=13$, $u=5$: $e'=2\neq6$); the objects of C3, C4 and C9 are invariant. -/86theorem p06010 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (u : Nat), (u % (4 : Nat) = (1 : Nat) → -(g ^ u) ^ κ = -g ^ κ) ∧ (u % (4 : Nat) = (3 : Nat) → -(g ^ u) ^ κ = - -g ^ κ) :=87  @FRC.Shell.Frame.orientation_class88/-- p06012 — 6:B5. $\Wt^{2}=-J$, $\Ft^{2}=J$, $\Ft^{4}=I$: the normalized operator generates the four-cycle, on the six shells of Table~\ref{tab:checks}. -/89theorem p06012 : (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (k j : Nat), FRC.Shell.sumRange (fun l => FRC.Shell.Frame.W g k l * FRC.Shell.Frame.W g l j) (p - (1 : Nat)) = -FRC.Shell.Frame.J (p - (1 : Nat)) k j) ∧ (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (k j : Nat), FRC.Shell.sumRange (fun l => FRC.Shell.Frame.Fmat g κ k l * FRC.Shell.Frame.Fmat g κ l j) (p - (1 : Nat)) = FRC.Shell.Frame.J (p - (1 : Nat)) k j) ∧ ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {k j : Nat}, k < p - (1 : Nat) → j < p - (1 : Nat) → FRC.Shell.sumRange (fun l => FRC.Shell.Frame.J (p - (1 : Nat)) k l * FRC.Shell.Frame.J (p - (1 : Nat)) l j) (p - (1 : Nat)) = if k = j then (1 : FRC.Shell p) else (0 : FRC.Shell p) :=90  And.intro @FRC.Shell.Frame.W_sq' (And.intro @FRC.Shell.Frame.F_sq (@FRC.Shell.Frame.J_sq))91/-- p06014 — 6:B7. $\Wt J=J\Wt$, hence $\Ft J=J\Ft$; every $v$ is uniquely $v^{+}+v^{-}$ with $Jv^{\pm}=\pm v^{\pm}$, a symmetric vector determined by its $2\kap+1$ entries at $0,\dots,2\kap$ and an antisymmetric one by its $2\kap-1$ entries at $1,\dots,2\kap-1$. -/92theorem p06014 : (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {k j : Nat}, k < p - (1 : Nat) → j < p - (1 : Nat) → FRC.Shell.sumRange (fun l => FRC.Shell.Frame.W g k l * FRC.Shell.Frame.J (p - (1 : Nat)) l j) (p - (1 : Nat)) = FRC.Shell.sumRange (fun l => FRC.Shell.Frame.J (p - (1 : Nat)) k l * FRC.Shell.Frame.W g l j) (p - (1 : Nat))) ∧ (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (v : Nat → FRC.Shell p), ∃ vp vm, FRC.Shell.Frame.Symm (p - (1 : Nat)) vp ∧ FRC.Shell.Frame.Antisymm (p - (1 : Nat)) vm ∧ ∀ (k : Nat), k < p - (1 : Nat) → v k = vp k + vm k) ∧ (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {a b : Nat → FRC.Shell p}, FRC.Shell.Frame.Symm (p - (1 : Nat)) a → FRC.Shell.Frame.Antisymm (p - (1 : Nat)) b → (∀ (k : Nat), k < p - (1 : Nat) → a k + b k = (0 : FRC.Shell p)) → ∀ (k : Nat), k < p - (1 : Nat) → a k = (0 : FRC.Shell p) ∧ b k = (0 : FRC.Shell p)) ∧ (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {v w : Nat → FRC.Shell p}, FRC.Shell.Frame.Symm (p - (1 : Nat)) v → FRC.Shell.Frame.Symm (p - (1 : Nat)) w → (∀ (k : Nat), k ≤ (2 : Nat) * κ → v k = w k) → ∀ (k : Nat), k < p - (1 : Nat) → v k = w k) ∧ ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {v w : Nat → FRC.Shell p}, FRC.Shell.Frame.Antisymm (p - (1 : Nat)) v → FRC.Shell.Frame.Antisymm (p - (1 : Nat)) w → (∀ (k : Nat), (0 : Nat) < k → k < (2 : Nat) * κ → v k = w k) → ∀ (k : Nat), k < p - (1 : Nat) → v k = w k :=93  And.intro @FRC.Shell.Frame.W_J_comm (And.intro @FRC.Shell.Frame.symm_antisymm_decomp (And.intro @FRC.Shell.Frame.symm_antisymm_unique (And.intro @FRC.Shell.Frame.symm_determined (@FRC.Shell.Frame.antisymm_determined))))94/-- p06027 — 6:D4. Scale-shift covariance: $S_r(M_m)=M_{m+r}$ for all $m$ and $r$, as ordered lists; consecutive entries of $M_m$ differ by the effective step $\gen^{m}$; $S_{r+(\p-1)}=S_r$, the periodicity of A5 in meridian form; at $\p=13$, $\gen=2$ the ladder $M_0,\dots,M_3$ at the steps $1,2,4,8$ (Section~\ref{sec:examples}). -/95theorem p06027 : (∀ {p : Nat} [FRC.Pos p] (g : FRC.Shell p) (κ m r : Nat), List.map (g.scale r) (g.meridian κ m) = g.meridian κ (m + r)) ∧ (∀ {p : Nat} [FRC.Pos p] (g : FRC.Shell p) (m a : Nat), FRC.Shell.ofNat (a + (1 : Nat)) * g ^ m = FRC.Shell.ofNat a * g ^ m + g ^ m) ∧ (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (r : Nat) (x : FRC.Shell p), g.scale (r + (p - (1 : Nat))) x = g.scale r x) ∧ FRC.Shell.meridian (2 : FRC.Shell (13 : Nat)) (3 : Nat) (0 : Nat) = [(0 : FRC.Shell (13 : Nat)), (1 : FRC.Shell (13 : Nat)), (2 : FRC.Shell (13 : Nat)), (3 : FRC.Shell (13 : Nat)), (4 : FRC.Shell (13 : Nat)), (5 : FRC.Shell (13 : Nat)), (6 : FRC.Shell (13 : Nat))] ∧ FRC.Shell.meridian (2 : FRC.Shell (13 : Nat)) (3 : Nat) (1 : Nat) = [(0 : FRC.Shell (13 : Nat)), (2 : FRC.Shell (13 : Nat)), (4 : FRC.Shell (13 : Nat)), (6 : FRC.Shell (13 : Nat)), (8 : FRC.Shell (13 : Nat)), (10 : FRC.Shell (13 : Nat)), (12 : FRC.Shell (13 : Nat))] ∧ FRC.Shell.meridian (2 : FRC.Shell (13 : Nat)) (3 : Nat) (2 : Nat) = [(0 : FRC.Shell (13 : Nat)), (4 : FRC.Shell (13 : Nat)), (8 : FRC.Shell (13 : Nat)), (12 : FRC.Shell (13 : Nat)), (3 : FRC.Shell (13 : Nat)), (7 : FRC.Shell (13 : Nat)), (11 : FRC.Shell (13 : Nat))] ∧ FRC.Shell.meridian (2 : FRC.Shell (13 : Nat)) (3 : Nat) (3 : Nat) = [(0 : FRC.Shell (13 : Nat)), (8 : FRC.Shell (13 : Nat)), (3 : FRC.Shell (13 : Nat)), (11 : FRC.Shell (13 : Nat)), (6 : FRC.Shell (13 : Nat)), (1 : FRC.Shell (13 : Nat)), (9 : FRC.Shell (13 : Nat))] ∧ (6 : Nat) * (2 : Nat) ^ (1 : Nat) < (13 : Nat) ∧ ¬(6 : Nat) * (2 : Nat) ^ (2 : Nat) < (13 : Nat) :=96  And.intro @FRC.Shell.meridian_scale (And.intro @FRC.Shell.meridian_step (And.intro @FRC.Shell.scale_periodic (@FRC.Shell.ladder13)))97/-- p08011 — 8:B5. The square class is chronon parity: the squares are $\langle g^{2}\rangle$, a residue's class is the parity of its drive-step count; on $\kap$-even shells every element of $Q_4=\{1,\im,-1,-\im\}$ is a square, so the order-four chart grading carries no signature. Exhaustive over every primitive root of $\F_{13}$, $\F_{17}$; Euler form on $\p<2000$. -/98theorem p08011 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (m : Nat), (∃ y, y * y = g ^ m) ↔ m % (2 : Nat) = (0 : Nat) :=99  @FRC.Shell.Frame.parity_iff100/-- p13044 — 13:H2. The exact angular carrier of $\pi$: $\chi(-1)=e^{\imR\piR}$ exactly in every shell, for every generator and either chirality, since $-1=\gen^{(\p-1)/2}$ sits at arc fraction one half --- Euler's identity as the half-turn tautology $\gen^{2\kap}=-1$ under the dictionary, with no calibration and no error term. -/101theorem p13044 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → g ^ ((2 : Nat) * κ) = (-1 : FRC.Shell p) :=102  @FRC.Shell.Frame.half_period103/-- p13051 — 13:I4. CRT stability: under the composite lift to $q=\prod\p_i$ the half-wall invariant of $\pi$ glues to $-2$ modulo every $q$, since $2\piA\equiv-1$ holds in every fibre, while the terminal content of $e$, $(-(\Ku{\p_i})^{-1})_i$, varies fibrewise. The two selectors resolve the $\p=13$ coincidence: the same residue $6$ reaches $\piR$ through C4 and $\eR$ through C1. -/104theorem p13051 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → FRC.Shell.ofNat ((2 : Nat) * FRC.Shell.Frame.halfPeriod κ) = (-1 : FRC.Shell p) :=105  @FRC.Shell.Frame.two_pi106/-- p13053 — 13:J2. Calibration pinning: the half-period satisfies the height-two relation $2\piA+1\equiv0$, the shell calibration $\p=4\kap+1$ itself, so the observer holds an exact short certificate of an entity it never accesses as a tally; the same pinning covers the quarter-turn ($\im^{2}+1\equiv0$) and the residue web of G2; the pin holds on every one of the $500$ shells $\p\le8009$ with $H(\piA)=2$. -/107theorem p13053 : (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → FRC.Shell.ofNat ((2 : Nat) * FRC.Shell.Frame.halfPeriod κ) = (-1 : FRC.Shell p)) ∧ ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → FRC.Shell.Frame.quarterTurn g κ * FRC.Shell.Frame.quarterTurn g κ = (-1 : FRC.Shell p) :=108  And.intro @FRC.Shell.Frame.two_pi (@FRC.Shell.Frame.quarter_turn_sq)109/-- p14031 — 14:X4. \textbf{The congruence $\dS$ even} acquires a second physical face: it is the existence of the octant sector, hence the well-definedness of the measured age --- a structure of the Carrier's cycle lattice, certified by realisation and read as an observational condition, never an arithmetic property of the numeral. -/110theorem p14031 : ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ((∃ r, r * r = (2 : FRC.Shell p)) ↔ ∃ m, κ = (2 : Nat) * m) :=111  @FRC.Shell.Frame.two_is_square_iff112/-- p26048 — 26:H3. Fourier-diagonalisable implies $\PC$: the additive transform on the shell is exact and self-inverse, $T^{-1}T=1$ on the full cycle $n=P-1$ (the round trips on the sub-cycles $n=8$, $256$, $2^{16}$ over $\F_{17}$, $\F_{257}$, $\F_{65537}$ [data]); the cyclic convolution costs $n^{2}$ multiplications in the position chart and $\tfrac32n\log_2n+n$ through the transform ($1\,048\,576$ against $16\,384$ at $n=2^{10}$) [data]. -/113theorem p26048 : (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ {k j : Nat}, k < p - (1 : Nat) → j < p - (1 : Nat) → FRC.Shell.sumRange (fun l => g ^ (l * k) * -g ^ ((p - (1 : Nat) - j) * l)) (p - (1 : Nat)) = if k = j then (1 : FRC.Shell p) else (0 : FRC.Shell p)) ∧ ∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (k : Nat), (0 : Nat) < k → k < p - (1 : Nat) → g ^ (p - (1 : Nat)) = (1 : FRC.Shell p) ∧ FRC.Shell.sumRange (fun j => (g ^ k) ^ j) (p - (1 : Nat)) = (0 : FRC.Shell p) ∧ FRC.Shell.ofNat (p - (1 : Nat)) = (-1 : FRC.Shell p) :=114  And.intro @FRC.Shell.Frame.dft_inverse (@FRC.Shell.Frame.principal_root)115-- end keys116117end FRC.Ledger118